<p>In this paper, we introduce variable Besov and Triebel-Lizorkin spaces via the semigroup generated by the Laguerre operator <i>L</i>, defined by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$P_{t,k}:=(t\sqrt{L})^ke^{-t\sqrt{L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>:</mo> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <msqrt> <mi>L</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>t</mi> <msqrt> <mi>L</mi> </msqrt> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k\in \mathbb {N}_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. Using the Calderón reproducing formula, molecular characterizations of these function spaces are established, leading to an embedding theorem and the boundedness for several classes of operators.</p>

Molecular Characterizations of Variable Besov and Triebel-Lizorkin Spaces Associated with Laguerre Operators

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发表日期：

2025年12月28日

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Yan Wang,
Fanghui Liao,
Zongguang Liu

摘要

In this paper, we introduce variable Besov and Triebel-Lizorkin spaces via the semigroup generated by the Laguerre operator L, defined by $P_{t,k}:=(t\sqrt{L})^ke^{-t\sqrt{L}}$ $P_{t, k} : = {(t \sqrt{L})}^{k} e^{- t \sqrt{L}}$ for $t>0$ $t > 0$ and $k\in \mathbb {N}_0$ $k \in N_{0}$ . Using the Calderón reproducing formula, molecular characterizations of these function spaces are established, leading to an embedding theorem and the boundedness for several classes of operators.